A Pretty Short Explanation of "Invert and Multiply"

I can't remember if I've seen this one before, but I thought it was worth sharing. I found it in a forthcoming book by Pamela Harris, at the end of an extended discussion about how to develop proportional reasoning.

We all have had students who know one rule for dividing fractions: invert and multiply. I learned it as "multiply by the reciprocal," while some of my students annoyed me with the vocabulary-loose "copy dot flip" (or flop). But why does it work?

There are many ways to explain it, including some that get into the gritty details of partitive and quotative division, but I think this explanation would satisfy most students and teachers. Let's start with a problem: \(\frac{3}{4} \div \frac{2}{3} \). I find this explanation is easiest to see if we write this as a compound fraction:
\[ \frac{\frac{3}{4}}{\frac{2}{3}} \]
The reasoning from here is simple: dividing by a fraction is hard, but dividing by one is really easy. But how do we turn \( \frac{2}{3} \) into one? By multiplying by it's reciprocal, of course. But we can't just multiply part of our problem by \( \frac{3}{2} \) without changing its value. The only thing we can multiply by without changing our value is one, and we can write one as something over itself, like this:
\[ \frac{\frac{3}{2}}{\frac{3}{2}} \]
So that's just a fancy way of writing one, and when we multiply the denominators we make a one:
\[ \frac{\frac{3}{4}}{\frac{2}{3}} \cdot \frac{\frac{3}{2}}{\frac{3}{2}} \rightarrow \frac{\frac{3}{4} \cdot \frac{3}{2}}{1} \rightarrow \frac{3}{4} \cdot \frac{3}{2} \]
Which explains "invert and multiply." It's just the result of wanting to divide by one instead of a fraction.